Inequalities on the Parameters of a Strongly Regular Graph

Abstract

In this paper we establish inequalities over the parameters and over the spectra of strongly regular graphs in the environment of Euclidean Jordan algebras. We consider a strongly regular graph, G, whose adjacency matrix A has three distinct eigenvalues, and the Euclidean Jordan algebra of real symmetric matrices of order n, $\text{Sym}(n, \mathbb{R})$ with the vector product and the inner product being the Jordan product and the usual trace of matrices, respectively. We associate a three dimensional real Euclidean Jordan subalgebra $\mathcal{A}$ of $\text{Sym}(n, \mathbb{R})$ to A, spanned by the identity matrix and the natural powers of A. Next, we compute the unique complete system of orthogonal idempotents $\mathcal{B}$ associated to A and we consider particular convergent Hadamard series constructed from the idempotents of $\mathcal{B}$. Finally, by the analysis of the spectra of the sums of these Hadamard series we establish new conditions for the existence of a strongly regular graph.